normality • normality

library(normality)

1. Overview

The normality R package provides a collection of statistical tools for assessing whether a sample is consistent with a normal distribution. It includes:

empirical distribution function (EDF)-based tests;
moment-based tests;
Shapiro-family tests;
skewness and kurtosis assessments;
high-level automatic normality assessment

2. Installation

Install the released version from CRAN:

install.packages("normality")

or the development version from GitHub:

if (!require("pak")) install.packages("pak")
pak::pak("P10911004-NPUST/normality")

Then load the package:

library(normality)

# For reproducibility
set.seed(123)

3. Quick start

The simplest way to assess a numeric vector is with is_normal().

x <- rnorm(30)
is_normal(x)
#> [1] TRUE

For a more detailed result:

is_normal(x, summary = TRUE)
#> $summary
#>                          alpha statistic      pval signif standard_value
#> Shapiro-Wilk-Royston (W)  0.05 0.9789351 0.7965839     ns     -0.8294810
#> D'Agostino-Pearson (K2)   0.05 0.4871878 0.7838059     ns      0.4871878
#> Anderson-Darling (A2)     0.05 0.1995093 0.8730283     ns      0.2049958
#>                          critical_value SE CI_lower CI_upper  N         AVG
#> Shapiro-Wilk-Royston (W)      1.6448536 NA       NA       NA 30 -0.04710376
#> D'Agostino-Pearson (K2)       5.9914645 NA       NA       NA 30 -0.04710376
#> Anderson-Darling (A2)         0.7307448 NA       NA       NA 30 -0.04710376
#>                                  MED       MIN      MAX        SD
#> Shapiro-Wilk-Royston (W) -0.07373326 -1.966617 1.786913 0.9810307
#> D'Agostino-Pearson (K2)  -0.07373326 -1.966617 1.786913 0.9810307
#> Anderson-Darling (A2)    -0.07373326 -1.966617 1.786913 0.9810307
#> 
#> $is_normal
#> [1] TRUE

If you only require a quick check for data normality, use is_normal and feel free to skip the remainder of this guide.

For explicit test selection, use check_normality():

check_normality(x, method = "SWR")
#>   x is_normal    Pvalue         W                                  method
#> 1 1      TRUE 0.7965839 0.9789351 Shapiro-Wilk-Royston (w) normality test

4. Understanding normality testing

Most formal normality tests use:

\[H_0: X \sim N(\mu, \sigma^2)\]

versus

\[H_1: X \not\sim N(\mu, \sigma^2)\]

A small p-value provides evidence against normality. A large p-value indicates that the test did not detect sufficient evidence of departure from normality.

A large p-value does not prove that the data are normally distributed.

Different tests are sensitive to different forms of departure. For example, skewness-based procedures focus on asymmetry, kurtosis-based procedures focus on tail weight, and EDF procedures compare the empirical and theoretical distributions.

5. Available tests

Method	Function	Sample-size
AD	`Anderson_Darling_test()`	≥ 8
CVM	`Cramer_von_Mises_test()`	≥ 8
LF	`Lilliefors_test()`	≥ 8
DAP	`D.Agostino_Pearson_test()`	≥ 20
JB	`Jarque_Bera_test()`	≥ 20
SW	`Shapiro_Wilk_test(method = "SW")`	3–50
SF	`Shapiro_Wilk_test(method = "SF")`	5–5000
SWR	`Shapiro_Wilk_test(method = "SWR")`	3–5000

5.1. Anderson–Darling test

The Anderson–Darling test is an EDF-based goodness-of-fit test that places greater emphasis on discrepancies in the tails.

ad <- Anderson_Darling_test(x)
#> 
#> ------------------------------------
#> Anderson-Darling (A2) normality test 
#> 
#> Statistic (A2) = 0.1995 
#> p-value = 0.87303
#> ------------------------------------

The summary table is available with:

ad$summary
#>                       alpha statistic      pval signif standard_value
#> Anderson-Darling (A2)  0.05 0.1995093 0.8730283     ns      0.2049958
#>                       critical_value SE CI_lower CI_upper  N         AVG
#> Anderson-Darling (A2)      0.7307448 NA       NA       NA 30 -0.04710376
#>                               MED       MIN      MAX        SD
#> Anderson-Darling (A2) -0.07373326 -1.966617 1.786913 0.9810307

5.2. Cramér–von Mises test

The Cramér–von Mises test is also EDF-based and evaluates discrepancies throughout the distribution.

cvm <- Cramer_von_Mises_test(x)
#> 
#> ------------------------------------
#> Cramer-von-Mises (W2) normality test 
#> 
#> Statistic (W2) = 0.0286 
#> p-value = 0.85958
#> ------------------------------------

Conceptually, Anderson–Darling emphasizes the tails more strongly, whereas Cramér–von Mises gives more uniform weight to discrepancies.

5.3. Lilliefors test

The Lilliefors test is an EDF-based procedure derived from the Kolmogorov–Smirnov framework and adapted for testing normality when the population mean and standard deviation are estimated from the sample.

lf <- Lilliefors_test(x)
#> 
#> --------------------------
#> Lilliefors normality test 
#> 
#> Statistic (D) = 0.0911 
#> p-value = 0.7531
#> --------------------------

5.4. D’Agostino–Pearson test

The D’Agostino–Pearson test is an omnibus moment-based test combining information from skewness and kurtosis.

dap <- D.Agostino_Pearson_test(x)
#> 
#> --------------------------------------
#> D'Agostino-Pearson (K2) normality test 
#> 
#> Alternative: two.sided 
#> 
#> Skewness = 0.1427 ;  p-value = 0.7094 
#> Kurtosis = 2.3627 ;  p-value = 0.55507 
#> 
#> Statistic (K2) = 0.4872 
#> p-value = 0.78381
#> --------------------------------------

The alternative argument can be used for component skewness and kurtosis assessments:

D.Agostino_Pearson_test(x, alternative = "two.sided")

5.5. Jarque–Bera test

The Jarque–Bera test is another moment-based omnibus normality test.

jb <- Jarque_Bera_test(x)
#> 
#> ------------------------------------
#> Jarque-Bera (JB) normality test 
#> 
#> Alternative: two.sided 
#> 
#> Skewness = 0.1356 ;  p-value = 0.72474 
#> Kurtosis = 2.2078 ;  p-value = 0.22604 
#> 
#> Statistic (JB) = 0.8765 
#> p-value = 0.64518
#> ------------------------------------

5.6. Shapiro-family tests

The Shapiro-family tests are generally the most powerful. However, they are limited by ties in the data and provide no numerical indication of the nature of non-normality as a by-product (D’Agostino, 1986; Section 9.5).

Shapiro_Wilk_test() provides three related procedures:

SW: Shapiro–Wilk;
SF: Shapiro–Francia;
SWR: Shapiro–Wilk with Royston’s modifications (default).

Original Shapiro–Wilk

sw <- Shapiro_Wilk_test(x, method = "SW")
#> 
#> ---------------------------------
#>  Shapiro-Wilk (W) normality test  
#> 
#>  Statistic (W) = 0.9756
#>  p-value = 0.7146 
#> ---------------------------------

Shapiro–Francia

sf <- Shapiro_Wilk_test(x, method = "SF")
#> 
#> -------------------------------------
#>  Shapiro-Francia (W') normality test  
#> 
#>  Statistic (W) = 0.986
#>  p-value = 0.90151 
#> -------------------------------------

Shapiro–Wilk with Royston’s modifications

# Equivalent to Shapiro_Wilk_test(x)
swr <- Shapiro_Wilk_test(x, method = "SWR")
#> 
#> -----------------------------------------
#>  Shapiro-Wilk-Royston (w) normality test  
#> 
#>  Statistic (W) = 0.9789
#>  p-value = 0.79658 
#> -----------------------------------------

Very large samples

The Shapiro-family procedures have finite sample-size limits in their direct implementations. Shapiro_Wilk_test() applies an re-sampling mechanism by default for observations above the maximum supported sample size.

x_large <- rnorm(10000)
Shapiro_Wilk_test(x_large, resampling = TRUE)

6. Testing multiple groups

The high-level functions support data frames and formulas.

data("iris")
check_normality(iris, Sepal.Length ~ Species, method = "SWR")
#>            x is_normal    Pvalue         W
#> 1     setosa      TRUE 0.4595132 0.9776985
#> 2 versicolor      TRUE 0.4647370 0.9778357
#> 3  virginica      TRUE 0.2583148 0.9711794
#>                                    method
#> 1 Shapiro-Wilk-Royston (w) normality test
#> 2 Shapiro-Wilk-Royston (w) normality test
#> 3 Shapiro-Wilk-Royston (w) normality test

The formula has the form:

response ~ group

The same interface is available in is_normal():

out <- is_normal(iris, Sepal.Length ~ Species, summary = TRUE)

For samples fewer than 8 observations, the automatic assessment relies on Shapiro–Wilk–Royston.

For samples from 8 through 19 observations, it combines:

Shapiro–Wilk–Royston;
skewness; and
Anderson–Darling.

For samples of at least 20 observations, it combines:

Shapiro–Wilk–Royston;
D’Agostino–Pearson; and
Anderson–Darling.

The sensitivity argument controls how many component assessments must support normality. Its allowed range is 1 to 3. A larger value makes the criterion more conservative.

set.seed(123)
x1 <- c(rnorm(20), runif(10))

for (i in 1:3)
{
    res <- is_normal(x1, sensitivity = i)
    print(sprintf("When sensitivity is %i: %s", i, res))
}
#> [1] "When sensitivity is 1: TRUE"
#> [1] "When sensitivity is 2: TRUE"
#> [1] "When sensitivity is 3: FALSE"

7. Skewness

The skewness() function assesses distributional asymmetry. Three estimators (G1, b1, g1) are available (Joanes & Gill, 1998, Wright & Herrington, 2011).

skewness(x, method = "G1")
#> 
#> --------------------------------------
#> Skewness test 
#> 
#> Alternative: two.sided 
#> 
#> Skewness (G1) = 0.1503
#> p-value = 0.72474
#> --------------------------------------
skewness(x, method = "b1")
#> 
#> --------------------------------------
#> Skewness test 
#> 
#> Alternative: two.sided 
#> 
#> Skewness (b1) = 0.1356
#> p-value = 0.72474
#> --------------------------------------
skewness(x, method = "g1")
#> 
#> --------------------------------------
#> Skewness test 
#> 
#> Alternative: two.sided 
#> 
#> Skewness (g1) = 0.1427
#> p-value = 0.72474
#> --------------------------------------

8. Kurtosis

The kurtosis() function assesses tail-related distributional characteristics. Three estimators (G2, b2, g2) are available (Joanes & Gill, 1998, Wright & Herrington, 2011).

kurtosis(x, method = "G2")
#> 
#> --------------------------------------
#> Kurtosis test 
#> 
#> Alternative: two.sided 
#> 
#> Kurtosis (G2) = 2.4723
#> p-value = 1.47373
#> --------------------------------------
kurtosis(x, method = "b2")
#> 
#> --------------------------------------
#> Kurtosis test 
#> 
#> Alternative: two.sided 
#> 
#> Kurtosis (b2) = 2.2078
#> p-value = 1.77396
#> --------------------------------------
kurtosis(x, method = "g2")
#> 
#> --------------------------------------
#> Kurtosis test 
#> 
#> Alternative: two.sided 
#> 
#> Kurtosis (g2) = 2.3627
#> p-value = 1.63723
#> --------------------------------------

9. Interpreting results

If a test produces a p-value of 0.12, the appropriate conclusion is:

The test did not provide sufficient evidence to reject the null hypothesis of normality.

If the p-value is 0.003:

The test provided evidence against the assumption of normality.

The next step should be to investigate the nature and practical importance of the departure.

Summary

The normality package provides a unified toolkit for assessing departures from normality.

The main functions are:

is_normal() for high-level automatic assessment;
check_normality() for selecting a specific normality test;
Shapiro_Wilk_test() for Shapiro-family procedures;
Anderson_Darling_test() for tail-sensitive EDF testing;
Cramer_von_Mises_test() for overall EDF discrepancy;
Lilliefors_test() for KS-type normality assessment;
D.Agostino_Pearson_test() and Jarque_Bera_test() for moment-based tests;
skewness() for distributional asymmetry; and
kurtosis() for tail-related characteristics.

Formal tests are most informative when considered together with sample size, graphical diagnostics, and the assumptions of the statistical method that will subsequently be applied.

References

Anderson, T.W., & Darling, D.A. (1954). A test of goodness of fit. Journal of the American Statistical Association, 49, 765–769. https://doi.org/10.1080/01621459.1954.10501232

D’Agostino, R.B. (1986). Goodness-of-Fit-Techniques (1st ed.). Routledge. https://doi.org/10.1201/9780203753064

D’Agostino, R.B., Belanger, A., & D’Agostino, R.B. (1990). A suggestion for using powerful and informative tests of normality. The American Statistician, 44, 316–321. https://doi.org/10.1080/00031305.1990.10475751

Jarque, C.M., & Bera, A.K. (1987). A test for normality of observations and regression residuals. International Statistical Review, 55, 163–172. https://doi.org/10.2307/1403192

Joanes, D.N., Gill, C.A. (1998). Comparing measures of sample skewness and kurtosis. Journal of the Royal Statistical Society: Series D (The Statistician), 47, 183–189. https://doi.org/10.1111/1467-9884.00122

Molin, P. & Abdi, H. (1998). New table and numerical approximations for Kolmogorov-Smirnov/Lilliefors/Van Soest normality test. Technical report, University of Bourgogne.

Royston, P. (1992). Approximating the Shapiro–Wilk W-test for non-normality. Statistics and Computing, 2, 117–119. https://doi.org/10.1007/BF01891203

Royston, P. (1993). A pocket-calculator algorithm for the Shapiro–Francia test for non-normality: an application to medicine. Statistics in Medicine, 12, 181–184. https://doi.org/10.1002/sim.4780120209

Shapiro, S.S. & Francia, R.S. (1972). An approximate analysis of variance test for normality. Journal of the American Statistical Association, 67, 215–216. https://doi.org/10.1080/01621459.1972.10481232

Shapiro, S.S. & Wilk, M.B. (1965). An analysis of variance test for normality (complete samples). Biometrika, 52, 591–611. https://doi.org/10.2307/2333709

Thode, H.C. (2002). Testing For Normality (1st ed.). CRC Press. https://doi.org/10.1201/9780203910894

Wright, D.B. & Herrington, J.A. (2011). Problematic standard errors and confidence intervals for skewness and kurtosis. Behavior Research Methods, 43, 8–17. https://doi.org/10.3758/s13428-010-0044-x